Oh this is not good.
If there are any eggs left that happen to be fertile, it might be a good idea to take out the chicks, and try to raise them themselves. They will have to find some baby bird food at a pet store, as well as a feeding syringe if they don't have one, already.
The parents should be feeding them every 3-4 hours. If they are not doing it that often, then take them out and feed them by hand. Sitting on the babies doesn't harm them though. This is how they keep them warm.
The parents could be depressed, if they are not aware of why their own chicks are dying. I mean, if I had a baby for the first time ever, and it died shortly after birth, I would be pretty depressed, too. They could also be sick, which could result in the loss of appetite.
Since they are first-time parents, they could be having trouble keeping up, and have started giving up. There are some cases where birds will kill their own babies, due to some kind of stress. Either it could be the stress of raising chicks, or something else is stressing them out.
If they need any, I can refer some guides on how to raise them. What IS important though, is that they stay warm when you take them out of the nest. At this age, they are very sensitive to cool temperatures, and will not survive if it's too cold. They could also check if it's cold, in the area of the cage, or if there's a draft. Then they could probably put a sheet on top. Not to completely cover, the cage, but at least the top and backside, to prevent from any drafts.
Also, how big is the cage? My mom says they may do this if the cage is too small, and that the parents don't think there is enough room for the chicks.